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PHYSICAL REVIEW LETTERS
PRL 115, 238101 (2015)
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Hydrodynamics Versus Intracellular Coupling in the Synchronization
of Eukaryotic Flagella
1
Greta Quaranta,1 Marie-Eve Aubin-Tam,2,* and Daniel Tam1,†
Laboratory for Aero and Hydrodynamics, Delft University of Technology, 2628CD Delft, Netherlands
2
Department of Bionanoscience, Delft University of Technology, 2628CJ Delft, Netherlands
(Received 30 June 2015; published 30 November 2015)
The influence of hydrodynamic forces on eukaryotic flagella synchronization is investigated by
triggering phase locking between a controlled external flow and the flagella of C. reinhardtii. Hydrodynamic forces required for synchronization are over an order of magnitude larger than hydrodynamic
forces experienced in physiological conditions. Our results suggest that synchronization is due instead to
coupling through cell internal fibers connecting the flagella. This conclusion is confirmed by observations
of the vfl3 mutant, with impaired mechanical connection between the flagella.
DOI: 10.1103/PhysRevLett.115.238101
PACS numbers: 87.16.Qp, 05.45.Xt, 47.63.Gd
The emergence of coherent behavior is ubiquitous in the
natural world and has long captivated physicists and
biologists alike [1]. Phase transitions leading to synchronization are observed between and within a variety of
biological organisms [2]. Recently, the organized dynamics
of micron sized hairlike cell projections called eukaryotic
flagella or cilia has attracted high levels of interest [3–5].
The ability of flagella to manipulate and transport fluid
relies on their capacity to spontaneously beat and synchronize with one another. Identifying the physical mechanisms leading to flagellar synchronization has been the
subject of intense investigations in recent years [6,7].
Theoretical studies on flagellar synchronization in
the inertialess viscous regime began with work by
Taylor [8], which highlighted the role of hydrodynamics.
Experimental work has focused on a model organism for
motility: unicellular green alga Chlamydomonas reinhardtii. C. reinhardtii possesses two flagella, the cis
and the trans flagellum, which beat in synchrony for
long time periods [7,9–11]. Flagella have often been
suggested to synchronize due to interflagellar hydrodynamic interactions. This view has been supported by
several theoretical studies [3,4,12,13] and recent experiments [14,15]. Recently, another view has emerged,
suggesting that the cell rocking motion causes synchronization by creating synchrony-restoring hydrodynamic
drag on the flagella [16,17].
Both mechanisms imply that synchronization is mediated by hydrodynamic forces. To what extent flagella
respond to hydrodynamic forces remains undetermined.
Here, we develop an experimental approach to actively
interact with C. reinhardtii. Controlled hydrodynamic
forces are applied on the cell by generating external
periodic background flows. Controlled perturbations have
been imposed on biological systems in previous investigations of hair cells and flagella [14,18–20]. Our experiments reveal that flagellar beating can be controlled solely
0031-9007=15=115(23)=238101(5)
via hydrodynamic forces generated by an external periodic
flow. However, we find the hydrodynamic forces required
for synchronization to be over an order of magnitude larger
than the forces experienced by the cell in physiological
conditions and, hence, unlikely to be relevant to synchronization in C. reinhardtii. Our results suggest instead that
coupling occurs through the cell-internal fiber connecting
the flagella. The potential role of such fibers in the
synchronized behavior of C. reinhardtii remains unclear.
These cell-internal fibers have been hypothesized to determine and modulate the flagella beating mode [9,21] and
have also been suggested to provide the mechanical
connection at the origin of synchronization itself
[22–24]. This last hypothesis is corroborated by our
experimental observations of the vfl3 mutant, with impaired
mechanical connection between the flagella.
Experimental approach.—Wild-type (wt) C. reinhardtii
(strain CC125) is grown in Tris-minimal medium [25] with
sterile air bubbling, and subject to light:dark (14:10 h)
cycles with light intensity of 230 μE · m−2 · s−1 . When
culture density reaches 106 cells · ml−1 , the suspension is
diluted in Tris media. C. reinhardtii mutant vfl3 (strain
CC1686) is grown similarly in TAP medium [25]. A
custom-made flow chamber of height h ¼ 1.5 mm, with
a 15 × 1.5 mm rectangular opening on one side, is filled
with the diluted cell suspension until the air-water interface
is pinned on all edges of the opening. The flow chamber is
placed on a piezoelectric stage (Nano-Drive, Mad City
Labs) under an inverted microscope with a 60× waterimmersion objective. A micropipette is inserted inside the
chamber, without direct contact with the chamber. A single
cell is captured at the tip of the micropipette; see Fig. 1.
When the stage position XðtÞ varies, the cell remains fixed
in the laboratory frame of reference. We impose XðtÞ as a
triangular wave of amplitude AF and frequency f F. The
motion of the piezo is calibrated in separate experiments by
tracking microbeads. The flow velocity field is measured
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© 2015 American Physical Society
PRL 115, 238101 (2015)
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PHYSICAL REVIEW LETTERS
(a)
(a)
(b)
(b)
(c)
(c)
FIG. 1 (color online). (a) Experimental setup. (b) Snapshots
representing the stroke patterns of C. reinhardtii, for a cell with
f0 ¼ 54.3 Hz and an external flow with fF ¼ 52.7 Hz and
U F ¼ 527 μm:s−1 . Time between snapshots: 6 ms. Arrows
represent the direction of the background flow. In this sequence,
the flagella beat with the flow and the flagellar tip motion has the
same direction as the external flow. (c) Snapshots of same cell as
in (b) 0.33 s later. The flagella beat against the flow.
inside the chamber around the micropipette and compared
with numerical computations of the velocity field around a
micropipette; see Supplemental Material [26]. We find the
motion of the stage to induce a unidirectional periodic
background flow around the cell with constant velocity
UðtÞ ¼ U F ¼ 2AF f F corresponding to a constant periodic forcing.
Flagella motion is recorded under bright field illumination of light intensity 160 μE · m−2 · s−1 with a sCMOS
camera (PCO.edge 5.5) at 838.4 fps. Movies are processed
to track the passage of each flagellum through an interrogation window [7,14]. From this procedure, we deduce
the phase dynamics of both the cis and the trans flagellum
and the time variations of the frequency spectrum, which
we represent in spectrograms; see Fig. 2(b) [31]. The phase
dynamics of the two flagella are nearly identical, with few
interflagellar slips recorded. We hence consider ϕðtÞ to
represent the phase of both beating flagella and deduce the
time-dependent phase difference ΔðtÞ between the flagella
and the external forcing flow ΔðtÞ ¼ ϕðtÞ − ϕF ðtÞ=2π,
where ϕF ðtÞ ¼ 2πf F t.
Experimental observations.—A cell is subject to a series
of different flow conditions. The flow amplitude AF remains
constant while the flow frequency f F is varied incrementally.
Recordings in the absence of flow are taken at regular
intervals to determine the intrinsic beating frequency of
the cell f 0 and monitor motility. Figures 2 and 3 present
experimental results for one particular cell with f 0 ¼
53.6 0.7 Hz. This cell is subject to background flows
of AF ¼ 5 μm and flow frequencies 49.9–60.3 Hz,
FIG. 2 (color online). Experimental data for one data set. The
same cell is subject to an external flow AF ¼ 5 μm and
fF ¼ 49.9–60.3 Hz. (a) Time variations of the phase difference
between the flagella and the external flow for separate recordings. Inset for f F ¼ 57.5 Hz shows a stepwise decay of ΔðtÞ.
Inset for f F ¼ 51.2 Hz shows fluctuations during phase locking. (b) Spectrograms of flagellar motion for different fF .
Black (white) correspond to a low (high) amplitude in the
frequency spectrum. From top: no synchrony (f F ¼ 60.3 Hz),
partial locking (fF ¼ 55.8 Hz), and complete phase locking
(fF ¼ 55.0 Hz). (c) f F − f as a function of f F − f 0 : Symbols,
experimental data; solid line, f is computed from Eq. (4) with
the fitted values for f 0, ϵ0 , and T eff ; dotted line, f is computed
from Eq. (4) in the limit of zero noise.
corresponding to flow velocities U F ¼ 499–603 μm · s−1 .
For f F ¼ 57.5–60.3 Hz, higher than the intrinsic frequency,
the phase difference ΔðtÞ decreases uniformly with time and
the flagella do not synchronize with the background flow,
Fig. 2(a). The beating frequency f remains close to f 0 and
does not lock with f F throughout the 30 s of recording.
Fluctuations around f 0 are observed in the beating frequency
[32] [Fig. 2(b), top]. The power and recovery strokes are
weakly affected by the direction of the flow and flagellar
strokes remain well defined regardless of whether the cell
beats with [Fig. 1(b)] or against the flow [Fig. 1(c)]. For the
lower value f F ¼ 57.5 Hz, ΔðtÞ decreases in stepwise
fashion, signifying the proximity of a synchronization
transition; see inset in Fig. 2(a). At f F ¼ 55.8 Hz, ΔðtÞ
is archetypical of a synchronization transition for a noisy
oscillator [33]. Periods of phase locking of duration up to 4 s
are interrupted by periods over which ΔðtÞ decreases as ϕðtÞ
progressively slips compared to ϕF ðtÞ, Fig. 2(a). When
the phases are locked, f ¼ f F , while f ¼ f 0 otherwise
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PHYSICAL REVIEW LETTERS
(a)
FIG. 3 (color online). Histograms showing the distributions of
phase difference ΔðtÞ in separate recordings of the same cell, with
external flows at AF ¼ 5 μm and different f F . Red line: P̂ðΔÞ
computed with the fitted parameters f0 ¼ 53.6 Hz,
ϵ0 =f 0 ¼ 0.042, and T eff =f 0 ¼ 0.0006.
[Fig. 2(b), center]. For f F ¼ 53.4–55 Hz, the phases remain
locked for the entire 30 s of each recordings in a 1:1 in-phase
locking mode with f ¼ f F [Fig. 2(b), bottom]. Few occasional phase slips are observed at f F ¼ 51.9 Hz, when the
cell performs one additional beat and Δ rapidly increases by
one unit. Finally, as f F is further decreased, the reverse
transition to asynchronous behavior takes place. Our observations are consistent with the behavior of a self-sustained
periodic oscillator under periodic external action in the
presence of small bounded noise [33]. The same experiments
were performed with the background flow in the direction
perpendicular to the pipette axis. In these experiments,
phase-locking was rarely triggered for such transverse flows.
Synchronization regions.—We proceed by determining
the hydrodynamic forces required to control flagellar
motility and achieve phase locking. We do so by fitting
our data to a low-order model for the synchronization of an
oscillator with an external forcing. The stochastic fluctuations of the phase difference ΔðtÞ are modeled as a
Langevin dynamics
dΔðtÞ
dVðΔÞ
¼−
þ ξðtÞ;
dt
dΔ
ð1Þ
where ξðtÞ represents a random noise in the flagellar
actuation and VðΔÞ is the potential associated with the
deterministic force driving ΔðtÞ. With −dVðΔÞ=dΔ ¼
−ν − ϵ sin½2πΔðtÞ, Eq. (1) yields the stochastic Adler
equation, where ν ¼ f F − f 0 is the detuning parameter
and ϵ the coupling strength between the flagella and the
external flow [33]. We assume ξðtÞ to be a Delta-correlated
Gaussian noise of intensity T eff, such that hξðτÞξðτ þ tÞi ¼
2T eff ΔðtÞ. From Eq. (1), one can derive a Fokker-Planck
equation for the probability density function PðΔ; tÞ:
(b)
FIG. 4 (color online). (a) Synchronization region in ðf F ; U F Þ
domain. Each marker represents a separate recording. U F ranges
from 86 μm · s−1 to 1300 μm · s−1 and is normalized with U 0 .
Color map represents the time fraction when flagellar beating is
phase locked with external flow. It is equal to 1 (black) when
phase locking is observed for the entire 30 s recording, while it is
0 (white) when phase locking is never observed. (b) Coupling
strength ϵ as a function of external flow U F =U 0.
∂P ∂½ν þ ε sinð2πΔÞP
∂ 2P
¼
þ T eff
.
ð2Þ
∂t
∂Δ
∂Δ2
The Fokker-Planck Eq. (2) has a time-independent solution
Z
1 Δþ1
VðΔ0 Þ − VðΔÞ
P̂ðΔÞ ¼
dΔ0 ;
exp
ð3Þ
C Δ
T eff
where C is a normalization constant [33]. The solution (3)
only depends on three parameters ν, ϵ, and T eff and is
computed numerically using the Gauss-Laguerre quadrature. For large jνj, P̂ðΔÞ is uniform. When jνj is decreased,
the probability P̂ðΔ ¼ 0Þ of in-phase beating increases,
while the probability P̂ðΔ ¼ 0.5Þ of antiphase beating
decreases. For small jνj, the phase is locked. P̂ðΔÞ is
Gaussian-like, with a zero probability to beat in antiphase;
see Fig. 3.
We examine cell motility over a wide range of amplitudes AF ¼ 0–10 μm and frequencies f F ¼ 43–68 Hz,
corresponding to UF ¼ 0–1360μm · s−1 . In comparison,
we measure the free swimming velocity to be U 0 ¼ 110
12μm · s−1 . Figure 4 presents our experimental data for 11
data sets and 171 separate recordings. Each data set
corresponds to recordings of the same cell subject to flows
of fixed AF and different f F . For low amplitude forcing,
UF ≈ 100 μm · s−1 , phase locking is never observed, regardless of how close f F is from f 0 . For U F ≥ 250 μm · s−1,
synchronization transitions similar to the ones in Figs. 2–3
are observed for all data sets.
The size of the synchronization region characterizes the
synchronization force and is determined by fitting experimental phase distributions with the solution (3) of Eq. (2).
P̂ðΔÞ depends on ν ¼ f F − f 0 , T eff , and ϵ. The noise level
T eff stems from stochasticity in the flagellar actuation and is
assumed to be a cell-dependent constant. We assume the
external forcing ϵ to scale directly with the hydrodynamic
force on the flagella and hence to be linear in the flow
velocity, ϵ ∼ U F . Within a data set, AF is constant and we
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write ϵ ¼ ϵ0 f F =f 0 . Hence, P̂ðΔÞ depends on f 0 , ϵ0 , and
T eff , which are determined by least-squares fitting. Figure 3
shows the agreement between the distribution of phase
differences for the data set detailed in Figs. 2–3 and the
computed P̂ðΔÞ. We compute the average beating frequency predicted by Eq. (1) as
Z
f¼
0
1
−
dV
P̂ðΔÞdΔ:
dΔ
ð4Þ
Experimental measurements of f agree with Eq. (4); see
Fig. 2(c). At the synchronization transition, the square root
behavior of f carries the characteristic signature of a
saddle-node bifurcation. These results confirm that the
stochastic phase dynamics is accurately represented with
only three parameters f 0 , T eff , and ϵ0 .
The fitting procedure is repeated for each data set and ϵ is
measured for a wide range of flow conditions. The values for
f 0 and T eff are consistent between cells with f 0 ¼ 52.6 1.1 Hz and T eff =f 0 ¼ 0.0008 0.0003. Our measurements
of T eff agree with previously reported values [10,34]. We
find ϵ to increase linearly with U F and directly measure ϵ as
a function of the hydrodynamic forces ϵ ¼ μU F =U0 with
μ ¼ 0.51 s−1 ; see Fig. 4(b). This result agrees with the shape
of an Arnold tongue, the prototypical synchronization region
of the model system (1); see Fig. 4(a).
Discussion.—Recent work established synchronization
through hydrodynamic interactions between two isolated
flagella of Volvox, whose intrinsic beating frequencies
differed by ∼10% [14]. Volvox and C. reinhardtii are
different organisms and the synchronization modes observed
in Ref. [14] are different from the symmetric breaststroke of
wt C. reinhardtii investigated here. Our experiments demonstrate that eukaryotic flagella respond to hydrodynamic
forces and can be synchronized with an external flow.
For C. reinhardtii, interflagellar synchronization requires
phase-locking between two flagella, whose intrinsic beating
frequencies differ by as much as 30%, hence, requiring
coupling strengths ϵ ≈ 15–20 Hz [35]. In contrast, in our
experiments for U F ≈ 10U 0, the hydrodynamic forces on the
flagella are an order of magnitude larger than those experienced for free-swimming cells, yet the flagella only
synchronize to flows with forcing frequencies within
5 Hz of f 0 . From Fig. 4(b), synchronization at 15–20 Hz
from f 0 would require flows of over 30U 0 ≈ 3300 μm:s−1.
Furthermore, for the largest flows U F ≈ 10U 0 when f F
is outside of the synchronization region, the velocity field
around the cell due to the background flow is 10–100 times
larger than the flow field due to hydrodynamic interactions,
see numerical calculations in the Supplemental Material
[26]. However, the flagella do not synchronize to the
background flow but do remain synchronized to each other
and perform symmetrical breaststrokes. In wt C. reinhardtii
performing breaststrokes, it seems unlikely that interflagellar synchronization is due to hydrodynamic interactions,
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since the velocity field is dominated by the background
flow to which the two flagella are not synchronized. Our
results suggest that interflagellar synchronization does not
primarly involve external hydrodynamic forces and point
instead towards a cell-internal synchronization mechanism.
Scanning electron micrographs (SEMs) of the region
around the basal bodies from which the flagella emerge
reveal that a ∼200 nm long contractile fiber, the distal
striated fiber, mechanically connects the basal bodies of the
two flagella [36]. It appears most plausible that this
mechanical connection through the distal striated fiber plays
a major role in flagellar synchronization for C. reinhardtii,
also discussed in Refs. [22,23]. Indeed, if we consider a force
balance on one flagellum, the total hydrodynamic force
exerted by the flagellum on the surrounding fluid is exactly
balanced by the direct mechanical force the same flagellum
exerts on the basal apparatus. As a result, strong mechanical
stress concentration is expected in the region of the cell
cortex around the basal apparatus of the flagella, which far
exceeds the viscous stresses inside the fluid. Synchronization
is therefore more likely mediated by elastic stresses, which
are conservative and act over an interflagellar distance of
only ∼200 nm in the distal fiber, rather than viscous stresses,
which are dissipative and act over an interflagellar distance
of ∼10 μm in the fluid.
To test this hypothesis, we investigated the mutant vfl3 of
C. reinhardtii. SEMs of vfl3 cells have revealed defects in
the distal striated fiber [24,37], which mechanically connects the flagella. Using the same experimental setup, we
observed the motility of 20 biflagellated vfl3 cells. Each
individual flagellum of vfl3 beats actively with normal
waveforms and frequencies. However, for this mutant with
impaired mechanical connection, the two flagella are
always observed to beat in asynchronous fashion for the
entire duration of the recordings, with no frequency locking
recorded in flagellar beating, except in one cell where
periods of phase locking were recorded. For most cells, the
two flagella beat in the same direction, in contrast with wt
cells, whose flagella beat in opposite direction when
performing a breaststroke; see also Ref. [24]. For vfl3
cells, we found the slower flagellum to beat at 48.6 8.8 Hz and the faster one at 63.1 6.9 Hz. These observations are consistent with a synchronization mechanism
relying on the distal striated fiber, since for the mutant vfl3
with an impaired mechanical connection between the
flagella, the cis and the trans flagellum beat at their own
intrinsic frequency.
These experiments for C. reinhardtii highlight the role of
elastic stresses in the cell cortex for flagellar synchronization
and have wide implications for ciliates. Contractile fibers
connecting flagella and cilia are found across organisms
[21]. In particular, for multiciliated cells, the network of
elastic actin fibers connecting dozens of cilia was reported
to play a role in coordination of ciliary beating [38,39].
Perturbing the bridges between these actin fibers resulted in
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loss of metachronal synchrony of cilia beating [38]. Further
investigation in other ciliated organisms is needed to assess
the prevalence of such intracellular coupling. The role of the
striated fibers is currently investigated for C. reinhardtii.
We thank Jerry Westerweel for useful conversations, and
Roland Kieffer, Vinesh Badloe, and Edwin Overmars for
their help in setting up the experiment. The work has been
supported by a TU Delft Technology Fellowship and a
Marie Curie CIG Grant (No. 618454).
*
m.e.aubin‑[email protected]
[email protected]
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